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Sections 7.1 and 7.2 This chapter presents the beginning of inferential statistics. The two major applications of inferential statistics Estimate a population parameter: proportion, mean Test some claim (or hypothesis) about a population. Point estimate: a single number Interval estimate: interval of numbers. Confidence Interval Why?: point estimate is not reliable under re-sampling. A confidence interval (CI): an interval of values used to estimate the true population parameter. Point Estimate p= population proportion ˆ= n p x sample proportion of x successes in a sample of size n. (pronounced Unbiased estimate (best estimate) ‘p-hat’) ˆ ˆ q = 1 - p = sample proportion of failures in a sample size of n Example: Photo-Cop Survey Responses 829 adult Minnesotans were surveyed, and 51% of them are opposed to the use of the photo-cop for issuing traffic tickets. Using these survey results, find the best estimate of the proportion of all adult Minnesotans opposed to photo- cop use. Best point estimate=sample proportion=51%. Confidence Level α: between 0 and 1 A confidence level: 1 - α or 100(1- α)%. E.g. 95%. This is the proportion of times that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. Other names: degree of confidence or the confidence coefficient. The Critical Value (z-score) Given α Finding zα/2 for 100(1- α)% Confidence Level α =5% α/2 = 2.5% = .025 Sampling Distribution of ^ p The sampling distribution of sample proportion can be approximated by a normal distribution if np≥15 and nq ≥15 : phat is approximately N(p, pq/n), q=1-p. p p− p ˆ z= ˆˆ pq n p ^ p ^ Margin of Error of p the maximum likely (with probability 1 – α) difference between the observed proportion ^ p and the true population proportion p. E = zα / 2 pˆ ˆq n ^ Standard Error of p =se Finding the 95% Confidence Interval for a Population Proportion A 95% confidence interval for a population proportion p is: ˆ ˆ p(1 - p) p ± 1.96(se), with se = ˆ n 100(1-α)% confidence interval for p is p(1 − p) ˆ ˆ p ± zα / 2 ( se) ˆ with se = n Example: Would You Pay Higher Prices to Protect the Environment? In 2000, the GSS asked: “Are you willing to pay much higher prices in order to protect the environment?” Of n = 1154 respondents, 518 were willing to do so Find and interpret a 95% confidence interval for the population proportion of adult Americans willing to do so at the time of the survey Example: Would You Pay Higher Prices to Protect the Environment? 518 p= ˆ = 0.45 1154 (0.45)(0.55) se = = 0.015 1154 E = 1.96(se) = 1.96(0.015) = 0.03 p ± E = 0.45 ± 0.03 = (0.42, 0.48) ˆ What is the Error Probability for the Confidence Interval Method? Summary: Effects of Confidence Level and Sample Size on Margin of Error The margin of error for a confidence interval: Increases as the confidence level increases Decreases as the sample size increases Determining Sample Size Recall : E= zα / 2 ˆˆ pq n (solve for n by algebra) n= zα / 2 p q ˆˆ 2 E2 Sample Size for Estimating Proportion p ˆ When an estimate p of p is known: n= ( zα / 2 )2 ˆˆ pq E2 When no estimate of p is known: ( zα / 2)2 0.25 n= E2 Example: Suppose a sociologist wants to determine the current percentage of U.S. households using e-mail. How many households must be surveyed in order to be 95% confident that the sample percentage is in error by no more than four percentage points? a) Use this result from an earlier study: In 1997, 16.9% of U.S. households used e-mail (based on data from The World Almanac and Book of Facts). b) Assume that we have no prior information suggesting a possible value of p. a) Use this result from an earlier study: In 1997, 16.9% of U.S. households used e-mail (based on data from The World Almanac and Book of Facts). n = [za/2 ]2ˆ ˆq p To be 95% confident that our E2 sample percentage is within four percentage points of the = [1.96]2 (0.169)(0.831) true percentage for all 0.042 households, we should randomly select and survey = 337.194 338 households. = 338 households b) Assume that we have no prior information suggesting a possible value of p. n = [za/2 ]2 • 0.25 E2 With no prior information, = (1.96)2 (0.25) we need a larger sample to 0.042 achieve the same results with 95% confidence and an = 600.25 error of no more than 4%. = 601 households Finding the Point Estimate and E from a Confidence Interval Point estimate of p: ˆ p = (upper confidence limit) + (lower confidence limit) 2 Margin of Error: E = (upper confidence limit) — (lower confidence limit) 2

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posted: | 7/8/2011 |

language: | English |

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